Is there any function to create strange alternating patterns of signs and ceros?

Asked Jan 28 '24 at 04:42

Active Jan 28 '24 at 06:16

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Hello there fellow mathematicians,

I'm trying to reduce a strange pattern of constants to a simpler way, but I find that the pattern of signs are changing as follows: (-, -, +, -, -, +, -, -, +, ...).

I tried a rare combination of the function floor to achieve that, but it is a little bit rough:

$(-1)^{ n \hspace{1mm}+ \left \lfloor\frac{n + 1}{3} \right \rfloor}$,

I couldn't find any simpler form to obtain the same result, so... Is there a easier way to express the pattern of signs previously said?

And in fact... I'm also looking for a way to get a similar pattern of 1's and 0's: (1, 1, 0, 1, 1, 0, 1, 1, 0, ...).

But for that I don't have the slightest idea how to achieve it, so... any suggestions?

edited Jan 28 '24 at 06:16

Gerry Myerson

185,413

asked Jan 28 '24 at 04:42

Carrot Carron't

1

For your pattern of $1,1,0,1,1,0,\cdots$, notice that it is your original pattern, but multiply by $-1$, add $1$, and take half. But as far as I know, there is no meaningful simplification beyond something of this type. Maybe the exponent could be made nicer, but that would be the extent of it. – PrincessEev Jan 28 '24 at 04:57
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Using mod: $n^2 \bmod 3$ – peterwhy Jan 28 '24 at 04:59
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Does this answer your question? General formula for nth element of the sequence 1, 1, 0, 1, 1, 0, 1, 1, 0, ... See also Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$ – Anne Bauval Jan 28 '24 at 05:36
@AnneBauval Oh, using trigonometric functions is a clever idea, thanks! – Carrot Carron't Jan 28 '24 at 05:46
Does this answer your question? Period 3: repeat [0, 1, 1]. – peterwhy Jan 28 '24 at 05:46

Is there any function to create strange alternating patterns of signs and ceros?

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